neural networks Balázs B Ujfalussy 17 october, 2016 idegrendszeri modellezés 2016 október 17.

Transcription

1 neural networks Balázs B Ujfalussy 17 october, 2016

2 Hierarchy of the nervous system behaviour idegrendszeri modellezés 1m CNS 10 cm systems 1 cm maps 1 mm networks 100 μm neurons 1 μm synapses 10 nm molecules 2016 október 17.

3 example II - Churchland et al neural dynamics underlying motor control

4 example II - Churchland et al neural dynamics underlying motor control large trial-to-trial variability internally generated activity with rich temporal structure!

5 external input networks - overview model neurons feed-forward connectivity recurrent connectivity the dynamics of the network defines the computation input computation response time input integration (working memory) many different dynamics - huge variety of computation time

6 how to model neural networks? bottom-up approach: include every detail where to stop? dendrites? stochastic channels? molecular dynamics? are we going to understand the model? - analysis is difficult! simulations are computationally expensive! how to fit the model s 1000s of parameters? top-down approach: analyse simple networks input is so strong that the detailed dynamics of individual neurons can be less important than the network level interaction between neurons what are the important features we want to reproduce? how to choose our simplified model? binary? firing rate? IF?

7 from spikes to rates neurons: implementing a mapping from input spike trains to a single output spike train: 1 (t) 2 (t) n (t) averaging argument: averaging over many independent neurons X N i (t) = apple X N Std i (t) X N (t) =F i (t) NX h i (t)i = / p N X N (t) F X N r (t) =F NX r i (t) / N r i (t) r i (t) (t) averaging over time would lead to the same result std is small compared to the mean what if inputs can have different sign?

8 firing rate dynamics: Wr : synaptic weight matrix symmetric (no self-connections) recurrent network models r dr dt = r + F (h + Wr) r = external input recurrent input 0 r 1 r 2... r n 1 C A non-linear recurrent networks: complex dynamics can perform difficult computations difficult to analyse linear recurrent networks: simple dynamics only trivial computations easy to analyse not realistic (e.g., negative rates) first step towards nonlinear

9 firing rate dynamics: Wr : synaptic weight matrix symmetric no self-connections linear recurrent network models X solution: eigenvalue decomposition c µ (t) =e T µ r(t) r dr dt = r + h + Wr external input We µ = µ e µ e T µ e = µ = recurrent input ( 1 ifµ = 0 otherwise r(t) = NX c µ (t) e µ eigenmodes evolve independently! µ=1 for normal matrices (loosely when W is symmetric), eigenvectors are orthogonal

10 firing rate dynamics: Wr : synaptic weight matrix symmetric µ = r = 1 µ no self-connections linear recurrent network models X dr r dt = solution: eigenvalue decomposition c µ (t) =e T µ r(t) 8 >< r + h + Wr µ < 0 µ < r! fast dynamics 0 < µ < 1 µ > r! slow dynamics >: 1 < µ µ < 0! unstable dynamics We µ = µ e µ e T µ e = µ = ( 1 ifµ = 0 otherwise r(t) = NX c µ (t) e µ eigenmodes evolve independently! µ=1 for normal matrices (loosely when W is symmetric), eigenvectors are orthogonal

11 the value of the eigenvalues determine the dynamics W = W11 W 21 W 22 W 12 dr 1 =0.7 r dt = r + h + Wr 2 =0.2 µ = r = 1 µ = 8 >< µ < 0 µ < r! fast dynamics 0 < µ < 1 µ > r! slow dynamics >: 1 < µ µ < 0! unstable dynamics c (t) c µ (t) =c µ (0) e t/( r/(1 µ)) connectivity causes slowing connected unconnected c (t) r (t) connected unconnected cell 1 cell 2 r (t) h (t) time (s) time (s)

12 the value of the eigenvalues determine the dynamics W = W11 W 21 W 22 W 12 dr 1 =0.7 r dt = r + h + Wr 2 =0.2 µ = r = 1 µ = 8 >< µ < 0 µ < r! fast dynamics 0 < µ < 1 µ > r! slow dynamics >: 1 < µ µ < 0! unstable dynamics c (t) c µ (t) =c µ (0) e t/( r/(1 µ)) connectivity causes slowing connected unconnected c (t) r (t) connected unconnected cell 1 cell 2 r (t) h (t) time (s) time (s)

13 the value of the eigenvalues determine the dynamics 1 = 1 dr r dt = r + h + Wr 2 = 2 µ = r c (t) W = W11 W 21 W 22 W 12 c µ (t) =c µ (0) e t/( r/(1 µ)) = 1 µ = 8 >< µ < 0 µ < r! fast dynamics 0 < µ < 1 µ > r! slow dynamics >: 1 < µ µ < 0! unstable dynamics c (t) r (t) connected unconnected r (t) connected unconnected cell 1 cell h (t) time (s) time (s)

14 homework problem W = W11 W 21 W 22 W 12 = p p 4/5 3/5 3/5 2/5 r dr dt = r + h + Wr Milyen aktivitást mutat a hálózat bemenetek hiányában tetszőleges kezdeti feltétel esetén, és hogyan reagál a külső bemenetekre? (4p) Miként változik meg a hálózat dinamikája, hogyha a súlymátrix jobb alsó elemét megváltoztatom (pl. W22 = 1/5 vagy W22 = 3/5)? (2p) Hogyan értelmeznéd ezeket az eredményeket? Milyen számításokat végéznek az egyes a neuronhálózatok? Milyen biológiai vagy komputációs problémát vetnek fel a megfigyelt aktivitásmintázatok? (4p)

15 symmetric, linear nets data c (t) r (t) connected unconnected time (s)

16 excitatory - inhibitory (EI) networks Previous assumption: symmetric W: Wij = W = Wji Dale s law: neurons are either excitatory or inhibitory WEE W IE W EI W II EI networks: Eigenvectors of non-normal matrices are not orthogonal! can NOT be the same eigenmodes do not give an intuitive picture about the dynamics of the network

17 W = example of a non-normal network W11 W 21 W 22 W 12 = r dr dt = r + h + Wr large initial eigenmodes c (t) 1 = = eigenmodes decay exponentially as before transient activity 1.5 connected unconnected r (t) time (s)

18 structured EI nets are able to generate rich transients

19 is the large variability of spikes consistent with the large number of synapses? Consistency argument: incoming and outgoing rates should be matched N ~ 10,000 incoming synapses: r XNX gr i ; E[r i ]=r; Var[r i ]= 2 NX r r, g X r i (t) O(1) g 1 N E[r ]= 1 N Var[r ]= 1 N 2 NX E[r i (t)] = r NX Var[r i (t)] = 1 N gets the mean but misses the variance 2 0

20 is the large variability of spikes consistent with the large number of synapses? Consistency argument: incoming and outgoing rates should be matched N ~ 10,000 incoming synapses: r XNX gr i ; E[r i ]=r; Var[r i ]= 2 NX r r, g X r i (t) O(1) g 1 N E[r ]= 1 N Var[r ]= 1 N 2 NX E[r i (t)] = r NX Var[r i (t)] = 1 N gets the mean but misses the variance 2 0 g 1 p N E[r ]= 1 p N X misses the mean but gets the variance right! Var[r ]= 1 N X N X E[r i (t)] = p Nr NX Var[r i (t)] 2

21 excitatory - inhibitory (EI) networks internally generated activity with rich temporal structure! large trial-to-trial variability excitatory sum inhibitory CV: std/mean, ~1 for a Poisson process

22 what we learned: activity dynamics in linear symmetric networks eigenvalues and eigenvectors rich transients in non-normal networks balance of excitation and inhibition is responsible for irregular firing what we missed: computation nonlinear networks spiking networks slides are based on David Barrett s slides (University of Cambridge) Dayan and Abbott: Theoretical Neuroscience (2001) Guillaume Hennequin s (U Cam) tutorial and papers (Hennequin et al., 2009) Murphy and Miller, 2009, Vogels et al., 2005